Mean Square Convergence Research Articles

A Fundamental Mean-Square Convergence Theorem for SDEs with Locally Lipschitz Coefficients and Its Applications

A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. Some numerical tests are presented.

Mean-square convergence of numerical approximations for a class of backward stochastic differential equations

This paper is devoted to the fundamental convergence theorem on the mean-square order of numerical approximations for a class of backward stochastic differential equations with terminal condition $\chi=\varphi(W_{T}+x)$. Our theorem shows that the mean-square order of convergence of a numerical method depends on the order of the one-step approximation for the mean-square deviation only. And some numerical schemes as examples are presented to verify the theorem.

On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids

We investigate the exponential long-time behaviour of the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise. Some results on the exponential convergence in mean square and with probability one of the weak probabilistic solution to the stationary solutions are given. We also prove an interesting result related to the stabilization of these stochastic evolution equations.

Optimal Output Tracking Control for Nonlinear Systems with External Disturbances Based on Stability Degree Constraint

This paper considers the optimal output tracking control problem for nonlinear systems affected by external disturbances based on stability degree constraint. The objective is to find an optimal output tracking controller, by which the cost function minimum and the state with the optimal having a higher mean-square convergence rate can be obtained. An optimal output tracking law is derived from a Riccati equation and Matrix equations. We give the existence and uniqueness conditions of the control law. Finally, a practical example is given to illustrate the effectiveness of the theory.

On the convergence of generalized polynomial chaos expansions

A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.

A note on order of convergence of numerical method for neutral stochastic functional differential equations

In this paper, we study the order of convergence of the Euler–Maruyama (EM) method for neutral stochastic functional differential equations (NSFDEs). Under the global Lipschitz condition, we show that the pth moment convergence of the EM numerical solutions for NSFDEs has order p/2 − 1/ l for any p ⩾ 2 and any integer l > 1. Moreover, we show the rate of the mean-square convergence of EM method under the local Lipschitz condition is 1 − ε/2 for any ε ∈ (0, 1), provided the local Lipschitz constants of the coefficients, valid on balls of radius j, are supposed not to grow faster than log j. This is significantly different from the case of stochastic differential equations where the order is 1/2.

Runge–Kutta methods for jump–diffusion differential equations

In this paper we consider Runge–Kutta methods for jump–diffusion differential equations. We present a study of their mean-square convergence properties for problems with multiplicative noise. We are concerned with two classes of Runge–Kutta methods. First, we analyse schemes where the drift is approximated by a Runge–Kutta ansatz and the diffusion and jump part by a Maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the Runge–Kutta ansatz for the drift. The second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise. We present results showing when the implicit stochastic equations defining the stage values of the Runge–Kutta methods are uniquely solvable. Finally, simulation results illustrate the theoretical findings.

Convergence rate of numerical solutions to SFDEs with jumps

In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the p th-moment convergence of Euler–Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1 / p for any p ≥ 2 . This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1 / 2 for any p ≥ 2 . It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1 / 2 , provided that local Lipschitz constants, valid on balls of radius j , do not grow faster than log j .

Kernel principal component analysis for stochastic input model generation

Stochastic analysis of random heterogeneous media provides useful information only if realistic input models of the material property variations are used. These input models are often constructed from a set of experimental samples of the underlying random field. To this end, the Karhunen–Loève (K–L) expansion, also known as principal component analysis (PCA), is the most popular model reduction method due to its uniform mean-square convergence. However, it only projects the samples onto an optimal linear subspace, which results in an unreasonable representation of the original data if they are non-linearly related to each other. In other words, it only preserves the first-order (mean) and second-order statistics (covariance) of a random field, which is insufficient for reproducing complex structures. This paper applies kernel principal component analysis (KPCA) to construct a reduced-order stochastic input model for the material property variation in heterogeneous media. KPCA can be considered as a nonlinear version of PCA. Through use of kernel functions, KPCA further enables the preservation of higher-order statistics of the random field, instead of just two-point statistics as in the standard Karhunen–Loève (K–L) expansion. Thus, this method can model non-Gaussian, non-stationary random fields. In this work, we also propose a new approach to solve the pre-image problem involved in KPCA. In addition, polynomial chaos (PC) expansion is used to represent the random coefficients in KPCA which provides a parametric stochastic input model. Thus, realizations, which are statistically consistent with the experimental data, can be generated in an efficient way. We showcase the methodology by constructing a low-dimensional stochastic input model to represent channelized permeability in porous media.

An averaging principle for stochastic dynamical systems with Lévy noise

The purpose of this paper is to establish an averaging principle for stochastic differential equations with non-Gaussian Lévy noise. The solutions to stochastic systems with Lévy noise can be approximated by solutions to averaged stochastic differential equations in the sense of both convergence in mean square and convergence in probability. The convergence order is also estimated in terms of noise intensity. Two examples are presented to demonstrate the applications of the averaging principle, and a numerical simulation is carried out to establish the good agreement.

Decentralized Parameter Estimation by Consensus Based Stochastic Approximation

In this paper, an algorithm for decentralized multi-agent estimation of parameters in linear discrete-time regression models is proposed in the form of a combination of local stochastic approximation algorithms and a global consensus strategy. An analysis of the asymptotic properties of the proposed algorithm is presented, taking into account both the multi-agent network structure and the probabilities of getting local measurements and implementing exchange of inter-agent messages. In the case of non-vanishing gains in the stochastic approximation algorithms, an asymptotic estimation error covariance matrix bound is defined as the solution of a Lyapunov-like matrix equation. In the case of asymptotically vanishing gains, the mean-square convergence is proved and the rate of convergence estimated. In the discussion, the problem of additive communication noise is treated in a methodologically consistent way. It is also demonstrated how the consensus scheme in the algorithm can contribute to the overall reduction of measurement noise influence. Some simulation results illustrate the obtained theoretical results.

Previously Unrecognized Links Between Statistical Dependence and Some Common Modes of Convergence

Most introductory courses in probability include the definitions of, and relationships between, convergence in distribution, probability, and mean square. Surprisingly, there are also links between the apparently unrelated notion of statistical dependence and two of these modes of convergence. These links are virtually absent from both the research literature and probability texts. Our goal here is to fill this gap, and by so doing provide some insight into the three modes of convergence. In particular, we establish ultimate dependence between members of a sequence of convergent random variables and its nondegenerate limit when convergence is in probability and in mean square. Through simple examples and graphical displays, we convey the heuristics that underlie the main results.

Numerical Approximations to the Stationary Solutions of Stochastic Differential Equations

In this paper, we investigate the possibility of approximating the stationary solution of a stochastic differential equation (SDE). We start with the random dynamical system generated by the SDE with the multiplicative noise. We prove that the pullback flow has a stationary point. However, the stationary point is not constructible explicitly; therefore, we look at the numerical approximation. We prove that the discrete time random dynamical system also has a stationary point. Finally, we prove mean-square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes, as well as almost surely convergence when the time step is rational.

Discrete Gradient Approach to Stochastic Differential Equations with a Conserved Quantity

General autonomous stochastic differential equations (SDEs) driven by one-dimensional Brownian motion in the Stratonovich sense with a conserved quantity $I(y)$ are considered. Relying on this conserved quantity, an equivalent “skew gradient” (SG) form of original SDEs is constructed. With the aim of constructing conserved numerical methods, direct discrete gradient approaches which approximate directly the SG system lead to two conserved methods. In indirect discrete gradient approaches, we first split the SG system into subsystems which preserve the original conserved quantity, then apply direct discrete gradient approaches to subsystems to produce two composition schemes. The mean-square convergence of order 1 for these four methods depends on the assumptions that the coefficients of SDEs are Lipschitz continuous with bounded second moments along the solution of SDEs, are twice continuous differentiable, and have derivatives that are uniformly bounded. Three numerical experiments are presented to verify our theoretical analysis and show the advantages of these methods.

Convergence and stability of the split-step θ-method for stochastic differential equations

In this paper, we construct a new split-step method for solving stochastic differential equations, namely the split-step θ -method. Under Lipschitz and linear growth conditions, we establish a mean-square convergence theory of split-step θ -approximate solutions. Moreover, the mean-square stability of the method for a linear test equation with real parameters is considered and the real mean-square stability region is plotted. Finally, numerical results are presented to demonstrate the efficiency of the split-step θ -method.

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.

Mean-Square Convergence Analysis of ADALINE Training With Minimum Error Entropy Criterion

Recently, the minimum error entropy (MEE) criterion has been used as an information theoretic alternative to traditional mean-square error criterion in supervised learning systems. MEE yields nonquadratic, nonconvex performance surface even for adaptive linear neuron (ADALINE) training, which complicates the theoretical analysis of the method. In this paper, we develop a unified approach for mean-square convergence analysis for ADALINE training under MEE criterion. The weight update equation is formulated in the form of block-data. Based on a block version of energy conservation relation, and under several assumptions, we carry out the mean-square convergence analysis of this class of adaptation algorithm, including mean-square stability, mean-square evolution (transient behavior) and the mean-square steady-state performance. Simulation experimental results agree with the theoretical predictions very well.

A Lax equivalence theorem for stochastic differential equations

In this paper, a stochastic mean square version of Lax’s equivalence theorem for Hilbert space valued stochastic differential equations with additive and multiplicative noise is proved. Definitions for consistency, stability, and convergence in mean square of an approximation of a stochastic differential equation are given and it is shown that these notions imply similar results as those known for approximations of deterministic partial differential equations. Examples show that the assumptions made are met by standard approximations.

Stochastic Runge–Kutta Methods for Itô SODEs with Small Noise

We consider stochastic Runge–Kutta methods for Itô stochastic ordinary differential equations, and study their mean-square convergence properties for problems with small multiplicative noise or additive noise. First we present schemes where the drift part is approximated by well-known methods for deterministic ordinary differential equations, and a Maruyama term is used to discretize the diffusion. Further, we suggest improving the discretization of the diffusion part by taking into account also mixed classical-stochastic integrals, and we present a suitable class of fully derivative-free methods. We show that the relation of the applied step-sizes to the smallness of the noise is essential to decide whether the new methods are worth the effort. Simulation results illustrate the theoretical findings.

Properties of FXLMS-Based Narrowband Active Noise Control With Online Secondary-Path Modeling

Rotating machines such as diesel engines, cutting machines, fans, motors, etc., generate sinusoidal noise signals that may be effectively reduced by narrowband active noise control (ANC) systems. In this paper, a typical filtered-X LMS (FXLMS) based narrowband ANC system equipped with an online secondary-path modeling subsystem is analyzed in detail. First, difference equations governing the dynamics of the FXLMS algorithm for secondary source synthesis and the LMS algorithm for secondary-path estimation are derived in terms of convergence in both mean and mean square. Steady-state expressions for mean-square error (MSE) as well as the residual noise power are then developed in closed form. Extensive simulations are performed to demonstrate the validity of the analytical results.